Prove that these relations on the set of all ISU students are equivalence relati
ID: 3566773 • Letter: P
Question
Prove that these relations on the set of all ISU students are equivalence relations. Describe the equivalence classes. Now, describe new equivalence relations which are refinements of the relations given.
(a) (a, b) is an element of R1 if and only if a and b live in the same dorm.
(b) (a, b) is an element of R2 if and only if a and b graduated from the same high school.
***PLEASE ANSWER COMPLETELY. "ISU" REPRESENTS SOME UNIVERSITY.
ALSO, BELOW IS A SCREENSHOT WITH THE PROBLEM FOR YOUR CONVENIENCE. THANKS***
Prove that these relations on the set of all ISU students are equivalence relations. Describe the equivalence classes. Now, describe new equivalence relations which are refinements of the relations given. (a) (a, b) is an element of R1 if and only if a and b live in the same dorm. (b) (a, b) is an element of R2 if and only if a and b graduated from the same high school.Explanation / Answer
(a)
Reflexivity:
(a,a) belongs to R4 as a lives in the same dorm as a.
Symmetric:
(a,b) belongs to R4 means a and b lived in the same dorm which implies b and a lived in the same dorm which means (b,a) belongs to R4.
Transvity:
(a,b) belongs to R4 and (b,c) belongs to R4 means a and b lived in the same dorm and b and c lived in the same dorm which implies a and c lived in the same dorm which means (a,c) belongs to R4.
(b)
Reflexivity:
(a,a) belongs to R5 as a graduated from the same high school as a.
Symmetric:
(a,b) belongs to R5 means a and b graduated from the same high school which implies b and a graduated from the same high school which means (b,a) belongs to R5.
Transvity:
(a,b) belongs to R4 and (b,c) belongs to R5 means a and b graduated from the same high school and b and c graduated from the same high school which implies a and c graduated from the same high school which means (a,c) belongs to R5.
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